1. **State the problem:** We are given the generating function $$G_X(s) = \frac{s^3}{6} (2 + s^4 + 2s^8)$$ and asked to find $$\text{mean}(3X + 5) + \text{Var}(5X + 7).$$ 2. **Recall formulas and properties:** - Mean of a linear transformation: $$E[aX + b] = aE[X] + b$$ - Variance of a linear transformation: $$\text{Var}(aX + b) = a^2 \text{Var}(X)$$ - Probability generating function (PGF): $$G_X(s) = E[s^X]$$ - First derivative at $$s=1$$ gives mean: $$G'_X(1) = E[X]$$ - Variance formula using PGF derivatives: $$\text{Var}(X) = G''_X(1) + G'_X(1) - (G'_X(1))^2$$ 3. **Calculate $$G_X(s)$$ explicitly:** $$G_X(s) = \frac{s^3}{6} (2 + s^4 + 2s^8) = \frac{1}{6} (2s^3 + s^7 + 2s^{11})$$ 4. **Find the first derivative $$G'_X(s)$$:** $$G'_X(s) = \frac{1}{6} (2 \cdot 3 s^2 + 7 s^6 + 2 \cdot 11 s^{10}) = \frac{1}{6} (6 s^2 + 7 s^6 + 22 s^{10})$$ 5. **Evaluate $$G'_X(1)$$ to find $$E[X]$$:** $$G'_X(1) = \frac{1}{6} (6 + 7 + 22) = \frac{35}{6}$$ 6. **Find the second derivative $$G''_X(s)$$:** $$G''_X(s) = \frac{1}{6} (6 \cdot 2 s + 7 \cdot 6 s^5 + 22 \cdot 10 s^9) = \frac{1}{6} (12 s + 42 s^5 + 220 s^9)$$ 7. **Evaluate $$G''_X(1)$$:** $$G''_X(1) = \frac{1}{6} (12 + 42 + 220) = \frac{274}{6} = \frac{137}{3}$$ 8. **Calculate variance $$\text{Var}(X)$$:** $$\text{Var}(X) = G''_X(1) + G'_X(1) - (G'_X(1))^2 = \frac{137}{3} + \frac{35}{6} - \left(\frac{35}{6}\right)^2$$ Calculate each term: $$\frac{137}{3} = \frac{274}{6}$$ So, $$\text{Var}(X) = \frac{274}{6} + \frac{35}{6} - \frac{1225}{36} = \frac{309}{6} - \frac{1225}{36}$$ Find common denominator 36: $$\frac{309}{6} = \frac{309 \times 6}{36} = \frac{1854}{36}$$ So, $$\text{Var}(X) = \frac{1854}{36} - \frac{1225}{36} = \frac{629}{36}$$ 9. **Calculate $$\text{mean}(3X + 5) + \text{Var}(5X + 7)$$:** $$E[3X + 5] = 3 E[X] + 5 = 3 \times \frac{35}{6} + 5 = \frac{105}{6} + 5 = \frac{105}{6} + \frac{30}{6} = \frac{135}{6} = \frac{45}{2}$$ $$\text{Var}(5X + 7) = 5^2 \text{Var}(X) = 25 \times \frac{629}{36} = \frac{15725}{36}$$ Sum: $$\frac{45}{2} + \frac{15725}{36} = \frac{45 \times 18}{36} + \frac{15725}{36} = \frac{810}{36} + \frac{15725}{36} = \frac{16535}{36}$$ **Final answer:** $$\boxed{\frac{16535}{36}}$$